David Cameron and Nick Clegg are wrangling over the membership and status of the inquiries that will be held into illegal phone hacking at the News of the World and wider questions about the future of media regulation.

The prime minister bowed to pressure to hold at least one inquiry but is resisting calls from Clegg for a judge to take charge.

The differences between Clegg and Cameron came as the government faced calls from across the Commons as well as from City shareholders to delay its final decision on the proposed takeover of BSkyB by News Corporation, parent company of the News of the World.

Jeremy Hunt, the culture secretary, gave the provisional go-ahead for the deal last Friday, subject to a final seven-day consultation over plans to hive off Sky News as a separately listed company to allay plurality fears. Hunt is due to consider thousands of pages of documents submitted during the consultation. He will then make a decision – which could be delayed into the summer recess – after consultations with Ofcom and the OFT.

The Labour leader, Ed Miliband, took the momentous step of turning against Rupert Murdoch's empire, calling for the resignation of News International's chief executive, Rebekah Brooks, and demanding the BSkyB decision be referred to the Competition Commission.

"The public will react with disbelief if next week the decision is taken to go ahead with this deal at a time when News International is subject to a major criminal investigation and we do not yet know who charges will be laid against," he said.

Simon Hughes, the Liberal Democrat deputy leader, said he would ask Ofcom to exercise its right to assess whether the directors of News Corp were "fit and proper" to take full control of BSkyB.

"Ofcom ... has a statutory obligation to consider at any time who is appropriate to hold a broadcasting licence. The message from this House must be that we want it actively to consider that obligation.

"If it comes to the view that the future owners of BSkyB are inappropriate, it should rule accordingly, which would mean that the BSkyB merger could not go ahead." Nicholas Soames, the former Tory defence minister, called for a pause in the BSkyB bid on the grounds of "serious criminality on the part of some people at News International". Soames is listened to with care because he is close to the Prince of Wales who was angered when Prince William's phone was hacked.

Several City shareholders called for Hunt to delay his final decision. Robert Talbut, chief investment officer of Royal London Asset Management said: "There are issues here that go beyond a simple financial transaction."

Another investor, speaking on the basis of anonymity said: "Hunt should hold fire because we are faced with a further concentration of media power in News Corp's hands at a time when there are allegations of malpractice within one of its major subsidiaries."

Shares in News Corp and BSkyB fell as the phone-hacking scandal put Murdoch and his bid to take control of the satellite broadcaster under scrutiny. News Corp shares fell by 5% at one stage on Wall Street, closing down 3.6% to $17.47. BSkyB shares in London fell 2.1% to 827p.

Murdoch denounced as "deplorable and unacceptable" the revelation in the Guardian earlier this week that the News of the World hacked into the telephone of Milly Dowler after she disappeared.

But he offered support for Brooks, who was NoW editor at the time of the schoolgirl's murder.

"I have made clear that our company must fully and proactively co-operate with the police in all investigations and that is exactly what News International has been doing and will continue to do under Rebekah Brooks's leadership."

It was reported last night that families of members of the armed forces killed in Afghanistan and Iraq may have been targeted by a private investigator who worked for the News of the World.

The Ministry of Defence was seeking clarification from Scotland Yard as sources said families of dead servicemen were being contacted by detectives over the scandal. News International said it would be "absolutely appalled and horrified" if there was any truth in the allegations and it would be immediately contacting the MoD.

MPH Solicitors – whose clients include Samantha Roberts, widow of one of the first Britons killed in Iraq in 2003 – called for clarity from authorities over the claims. Solicitor Geraldine McCool said the firm had been contacted by media yesterday over the allegations surrounding high-profile military inquests in 2006 and 2007. "We are making efforts to verify this information," a statement on the firm's website said.

The chancellor, George Osborne, was also notified by the Metropolitan police that his name and home phone number appeared in notes kept by Glenn Mulcaire and Clive Goodman, the private investigator and News of the World reporter who were both jailed in 2007 for phone hacking.

Cameron announced the inquiry or inquiries would be held after consulting Clegg. It is expected one inquiry will examine how phone hacking was started and tolerated, while a second will examine the future of media regulation.

But there were differences with Clegg over whether a judge would be involved. A Downing Street source said: "We do not have to have a judge-led inquiry to make it effective."

Clegg insisted a judge would have to be involved in at least one inquiry. In an email to Lib Dem members, he said: "The inquiries must be independent, open, able to access all information and call witnesses, and that crucially the inquiry dealing with legal issues (eg relationship between police and media) must be presided over by a judge."

One Lib Dem source said: "There is no point in having an enquiry if it does not have teeth. We do not want a talking shop. Unless you have a judge you can't deal with the crunchy bit."

The scale of the anger at News International across the Commons was highlighted when Tom Watson, a former Labour minister, accused it of entering the "criminal underworld" by "paying people to interfere with police officers and were doing so on behalf of known criminals".

He said James Murdoch, the tycoon's son, had "personally, without board approval, authorised money to be paid by his company to silence people who had been hacked and to cover up criminal behaviour within his organisation".

John Whittingdale, the Tory chairman of the Commons culture select committee, named a series of News International executives who had told his committee that only one reporter was responsible for phone hacking while police possessed evidence of widespread illegality.

News International is planning to relieve the pressure on Brooks, by claiming she was on holiday when a mobile phone belonging to Dowler was hacked into.

The Guardian understands that the company has established that Brooks, News of the World editor from May 2000 until January 2003, was on holiday in Italy when the paper ran a story which referred to a message that had been left on the teenager's phone.

That is likely to focus attention on Andy Coulson, who was Brooks's deputy at the time, and would normally have edited the paper in her absence.

Procter & Gamble, Britain's biggest advertiser, plus O2, Vauxhall, Butlins and Virgin Holidays, joined Ford in pulling ads from this weekend's News of the World. P&G spent almost £1.5m in the News of the World in the last year.